turkmath.org

Let $M$ be an ordered vector space over an ordered division ring $D$. A subset $X$ of $M^n$ is called "semilinear" if it is a boolean combination of sets defined by linear equations and inequalities with coefficients from $D$. A "semilinear group" is a group whose domain and the graph of its multiplication are semilinear sets. We prove that every semilinear group is semilinearly isomorphic to a quotient by a lattice, exemplifying a strong connection to real Lie groups. We then investigate analogous theorems for semialgebraic groups, as well as for other groups definable in o-minimal structures.